The Willmore flow near spheres

By Gieri Simonett. Differential and Integral Equations, 2001

137 citations recorded by the source corpus.

The Willmore ∞ow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of so- lutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/1c6a195c9bfa1d3acaacfd75cab74ca631736c33

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Smooth geometric evolutions of hypersurfaces (2001, background): “See [KS1,2], [ Si ] for the recent results on this subject.”
Error analysis of a finite element method for the Willmore flow of graphs (2006, background): “In [17] it is shown that a unique local solution of (1.”
The Wulff shape minimizes an anisotropic Willmore functional (2004, background): “is able to show global existence of the flow and convergence to a sphere [19].”
Gradient flow for the Willmore functional (2002, background): “For the Willmore flow Simonett shows long term existence for initial data which are close in C(2)' to a sphere; furthermore the solution gets spherical as t —> oo [11].”
Analysis aspects of Willmore surfaces (2008, methodology): “ii) Our approach should be very useful in the study of the Willmore Flow initiated in the work of Kuwert and Sch¨atzle [KS1], [KS2],[KS3] (see a lso [Sim]).”
The Willmore Flow with Small Initial Energy (2001, methodology): “1 The statement of the theorem was recently proved in [9] under the stronger assumption that f0 is close to a round sphere in the C2,αtopology, using a center manifold analysis which gives related stability results for a couple of other flows; see [2] for an overview.”
A level set formulation for Willmore flow (2004, background): “where h is the mean curvature on M, i. e., h is the sum of the principle curvatures on M. The corresponding L2-gradient flow ‐ the Willmore flow ‐ is given by the geometric evolution problem [34, 32 , 21]”
Computation of geometric partial differential equations and mean curvature flow (2005, background): “In Simonett (2001) it is shown that a unique local solution of (9.32) satisfying Γ(0) = Γ0 exists provided that Γ0 is a compact closed immersed and orientable C2,α-surface in R3.”